Two different quadratic equations have a common root. Let the three unique roots of the two equations be A, B and C - all of them are positive integers. If (A + B + C) = 41 and the product of the roots of one of the equations is 35, which of the following options is definitely correct?

It has been given that A+B+C = 41.
Let the common root be B. 
All the roots are positive integers. 
The product of the roots of one of the equations is 35. 
35 can be obtained only in 2 ways - either as 5*7 or 35*1. 
A+B+C = 41.
If A and B are 5 and 7 in any order, then C = 41 - 5 - 7 = 29.
If A and B are 35 and 1 in any order, then C = 41 - 35 - 1 = 5. 
As we can see, in either case, 5 is one of the 3 roots. 
Therefore, option C is the right answer.